Nobel Prize 2008 - broken symmetrieshep.fuw.edu.pl/seminars/Seminaria_2008-2009/Nobel2008.pdf · at a conference at Perdue (1960) reprinted in ”Broken Symmetries, Selected Papers

Nobel Prize 2008- broken symmetries -

Bohdan GRZADKOWSKI

University of Warsaw

• Nobel Prize Laureates in Physics 2008

• Yoichiro Nambu and spontaneous chiral symmetry breaking

• The Goldstone theorem

• Makoto Kobayashi and Toshihide Maskawa and explicit CP violation

• Comments:

– Giovanni Jona-Lasinio - ”the Nambu-Jona-Lasinio model”

– Jeffrey Goldstone - ”the Nambu-Goldstone bosons”

– Robert Brout, Francois Englert, Peter Higgs, . . . - the Higgs mechanism

– Nicola Cabibbo - the Cabibbo angle

• Cosmological illustrations e.g. baryogenesis

• We are not yet done: the strong CP problem

• Hints for beyond the Standard Model physics

• Conclusions

Nobel Prize 2008, December 5th, 2008, University of Warsaw 1

Nobel Prize Laureates in Physics 2008

• ”for the discovery of the mechanism of spontaneous broken symmetry in subatomicphysics”:

– Yoichiro Nambu, Enrico Fermi Institute, University of Chicago Chicago, IL,USA, 1/2 of the prize, born 1921,”A ”Superconductor” Model of Elementary Particles and its Consequences” givenat a conference at Perdue (1960) reprinted in ”Broken Symmetries, SelectedPapers by Y.Nambu” ed. T. Eguchi and K. Nishijima, World Scientific 1995

• ”for the discovery of the origin of the broken symmetry which predicts the existenceof at least three families of quarks in nature”:

– Makoto Kobayashi, High Energy Accelerator Research Organization (KEK)Tsukuba, Japan, 1/4 of the prize, born 1944

– Toshihide Maskawa, Kyoto Sangyo University; Yukawa Institute for TheoreticalPhysics (YITP), Kyoto University Kyoto, Japan, 1/4 of the prize, born 1940M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory ofWeak Interaction”, Prog. Theor. Phys. 49, 652 (1973), 5503 citations


Yoichiro Nambu and spontaneous chiral symmetry breaking

The state of the art in strong interactions in 1960:

• Isospin symmetry, SU(2) (introduced by Werner Heisenberg in 1932 to explainproperties of the then newly discovered neutron):

– mp = 938.3 MeV and mn = 939.6 MeV.– The strength of the strong interaction between any pair of nucleons is the same,

independent of whether they are interacting as protons or as neutrons.

• The baryon number conservation ↔ U(1)B symmetry

• 3 pions: π±, π0, with mπ± = 139.6 MeV and mπ0 = 135.0 MeV, somπ± ' mπ0 � mp ' mn

⇓The hypothesis of spontaneous chiral symmetry breaking

Nambu 1960, Goldstone 1961


• symmetry: SU(2)A × SU(2)V with Qa5 and Qa (a = 1, 2, 3) are generators of

SU(2)A,V

• vacuum |0〉: 〈0|H|0〉 = min〈H〉

The hypothesis of spontaneous chiral symmetry breaking: Qa5|0〉 6= 0 and Qa|0〉 = 0

Nambu 1960, Goldstone 1961

SU(2)A is spontaneously broken

⇓The theory contains massless bosons corresponding to each generator that does notannihilate the vacuum (since [Qa

5,H] = 0 therefore Qa5|0〉 6= 0 is degenerate with

|0〉 applying successively Qa5 one can generate infinite number of degenerate states -

massless particle).


The sigma model of the strong interactionsThe theory of pions as Nambu-Goldston bosons

-Gell-Mann & Levy-

The symmetry: SU(2)L × SU(2)R ⇔ SU(2)A × SU(2)V

The elementary fields:

N =(pn

)and ~π (0−), σ′ (0+)

The Lagrangian:

L = Ni 6∂N−gN (σ′ + i~σ · ~πγ5)N+12

[(∂µ~π)2 + (∂µσ

′)2]−1

2µ2(σ′2+~π2)−1

4λ(σ′2+~π2)2

where σ′ is needed to make L chirally symmetric, ~σ = (σ1, σ2, σ3) and

Ni 6∂N = NRi 6∂NR + NLi 6∂NL

N (σ′ + i~σ · ~πγ5)N = NL (σ′ + i~σ · ~π)NR + NR (σ′ − i~σ · ~π)NL

The transformation properties of σ′± i~σ · ~π are determined by the requirement of theinvariance of L under SU(2)L × SU(2)R:


• SU(2)L: σ′ → σ′ + 12~π · δ~αL and ~π → ~π − 1

2~π × δ~αL − 12σ′δ~αL

• SU(2)R: σ′ → σ′ − 12~π · δ~αR and ~π → ~π − 1

2~π × δ~αR + 12σ′δ~αR

The ground state is (at the tree level) the minimum of the potential:

V (σ′, ~π) =14λ

(σ′

2 + ~π2 +µ2

λ

)2

So, either

• σ′ = ~π = 0 for µ2/λ > 0, or

• σ′2 + ~π2 =∣∣∣µ2

λ

∣∣∣ for µ2/λ < 0

To satisfy Qa5|0〉 6= 0 and Qa|0〉 = 0 we must choose

〈0|~π|0〉 = 0 and 〈0|σ′|0〉 =(∣∣∣∣µ2

λ

∣∣∣∣)1/2

≡ v

Expanding around v, σ = σ′ − v one obtains:


L = N(i 6∂ − gv)N − gN (σ + i~σ · ~πγ5)N+

+12

[(∂µ~π)2 + (∂µσ)2

]− |µ2|σ2 − 1

4λ(σ2 + ~π2)2 − λv(σ3 + σ~π2) + const.

• So, we have massive nucleons mN = gv, a sigma meson of mass m2σ = 2|µ2| and

the isospin triplet of massless pions: (π+, π0, π−).

• Using the current implied by SU(2)A one finds 〈0|JaA µ(x)|πa(q)〉 = −〈0|σ′|0〉iqµδabe−iqx,

sofπ = −〈0|σ′|0〉

• since in reality mπ 6= 0, one should break the chiral symmetry explicitly preservingisospin SU(2): δL = −m3σ′. Then

m2π =

m3

fπ6= 0

• The axial current is not conserved and one obtains PCAC (Partial Conservation ofAxial Current)

∂µJaA µ(x) = m2

πfππa(x)


The Goldstone theorem

L =12∂µφi∂

µφi − V (φi)

If L is symmetric with respect to φi → φ′i ' φi − iΘaT aijφj then

0 = δV (φi) =∂V

∂φiδφi = −i∂V

∂φiΘaT a

ijφj (1)

Expanding V (φi) around its minimum φi = vi one gets (φ′i = φi − vi)

V (φi) = const. +12M2

ijφ′iφ′j + · · ·

where

M2ij =

(∂2V

∂φi∂φj

)φi=vi

Differentiating (1) at the minimum one gets

M2ij T

aijvj = 0 (2)

So, for all broken generators (T aijvj 6= 0), (2) implies the existence of a massless

Nambu-Goldstone boson.


Makoto Kobayashi and Toshihide Maskawa and explicit CP violation

The state of the art in quark weak interactions in 1973 :

• James Cronin, Val Fitch, 1964 =⇒ CP violation in K → ππ

• Steven Weinberg, ”A Model of Leptons”, Phys.Rev.Lett. 19, 1264 (1967)

• Charm quark predicted in 1970 by S. Glashow, J. Iliopoulos, and L. Maiani, Phys.Rev. D2, 1285 (1970)

• Quarks: u,d,s known while charm (c) still unobserved till its discovery in 1974

Kobayashi-Maskawa: N ≥ 3 implies CP violation in the Weinberg model


• Parity:

(t, ~x) P→ (t,−~x)S(t, ~x) P→ +S(t,−~x) scalarP (t, ~x) P→ −P (t,−~x) pseudoscalar

ψa(t, ~x)γµψb(t, ~x)P→ +ψa(t,−~x)γµψb(t,−~x) vector current

ψa(t, ~x)γµγ5ψb(t, ~x)P→ −ψa(t,−~x)γµγ5ψb(t,−~x) axial current

Vµ(t, ~x) P→ +V µ(t,−~x) vectorAµ(t, ~x) P→ −Aµ(t,−~x) axial

• Charge conjugation:

φC→ φ†

ψaγµψbC→ −ψbγµψa vector current

ψaγµγ5ψbC→ +ψbγµγ5ψa vector current

VµC→ −V ?

µ vector

AµC→ +A?

µ axial

where C−1γµC = −γTµ and e.g. C = iγ2γ0.


• CP :

(t, ~x) CP→ (t,−~x)φ

CP→ φ†

ψaγµψbCP→ −ψbγ

µψa vector currentψaγµγ5ψb

CP→ −ψbγµγ5ψa vector current

VµCP→ −V µ ? vector

AµCP→ −Aµ ? axial

The Standard Model in the interaction basis:

• gauge boson self-interactions symmetric under CP ,

• gauge boson ↔ fermion interactions symmetric under CP ,

• gauge boson ↔ Higgs boson interactions symmetric under CP ,

• Higgs boson self-interactions symmetric under CP ,

• Higgs boson ↔ fermion (Yukawa) interactions ??? under CP .


Hadronic Yukawa interactions in the SM

LY = −N∑

j,k=1

{[Γu

jk(u′, d′)j LHu′k R + Γu ?

jk u′k RH

†(u′

d′

)j L

][Γd

jk(u′, d′)j LHd′k R + Γd ?

jk d′k RH

†(u′

d′

)j L

]}

where

H =(φ+

φ0

)and H ≡ iσ2H

? =(

φ0 ?

−φ−)

(u′, d′)j LHu′k R

CP→ u′k RH†(u′

d′

)j L

(u′, d′)j LHd′k R

CP→ d′k RH†(u′

d′

)j L

If Γujk 6= Γu ?

jk and/or Γdjk 6= Γd ?

jk then CP invariance seems to be broken explicitly bythe Yukawa interactions !


Spontaneous symmetry breaking =⇒ H → 1√2

(0

v + h

)⇓

LY → Lm = −N∑

j,k=1

u′j L

[v√2Γu

jk

]︸︷︷︸

Mujk

u′k R + d′j L

[v√2Γd

jk

]︸︷︷︸

Mdjk

d′k R + H.c.

Mass matrix diagonalization (optional):

uL = ULu′L dL = DLd

′L for Mu = ULMuU†R diagonal

uR = URu′R dR = DRd

′R for Md = DLMdD†

R diagonal

LY = −N∑

j=1

[uLM

uuR + dLMddR + H.c.

](1 +

h

v

)= −

N∑j=1

[uMuu+ dMdd

](1 +

h

v

)W+

µ u′Lγ

µd′L+H.c. = W+µ uLγ

µULDL︸︷︷︸VCKM

dL+H.c. = W+µ uLγ

µVCKMdL+W−µdLγµV†CKMuL

W+µ

CP↔ −W−µ and uj Lγµdk L

CP↔ −dk Lγµuj L

VCKM = V ?CKM =⇒ CP conservation


Rephasing for N = 3: uL

cLtL

→

eiαu

eiαc

eiαt

uL

cLtL

and

dL

sL

bL

→

eiαd

eiαs

eiαb

dL

sL

bL

Then

VCKM →

e−iαu

e−iαc

e−iαt

VCKM

eiαd

eiαs

eiαb

so

V ijCKM → e−i(αj−αj)V ij

CKM

2N−1 phases could be removed (as unphysical), so eventually the number of relevantphases

N2 − N(N − 1)2

− (2N − 1) =(N − 1)(N − 2)

2CP violation appears for N ≥ 3


Comments

• Spontaneous symmetry breaking:

– 1960: Yoichiro Nambu was 39 years old, Giovanni Jona-Lasinio, who presentedNambu’s work at Perdue was 28 years old.

– Jeffrey Goldstone - ”the Nambu-Goldstone bosons”:Jeffrey Goldstone, “Field Theories With Superconductor Solutions”,Nuovo Cim. 19, 154 (1961).

– Robert Brout, Francois Englert and Peter Higgs - the Higgs mechanism:Awarded the High Energy and Particle Prize of the European Physical Societyin 1997 and the Wolf Prize in Physics in 2004 for the ”Higgs mechanism” (themechanism that generates mass for gauge vector bosons).

– Gerald Guralnik, Thomas Kibble, Carl Hagen - the Higgs mechanism, but noprize.

• CP violation:

– Nicola Cabibbo - the Cabibbo angle, no CP violation.


Cosmological illustrations e.g. baryogenesis

Baryogenesis is where CP violation and spontaneous symmetry breaking merge.

η ≡ nb − nb

nγ' nb

nγ' (1− 6)× 10−10

The three necessary ”Sakharov conditions” are:

1. Baryon number B violation.

2. C-symmetry and CP -symmetry violation.

3. Interactions out of thermal equilibrium.


Electroweak baryogenesis:

• ”CP -symmetry violation: Jarlskog invariant, Im(VudVcbV?ubV

?cd) from the Kobayashi-

Maskawa phase,

• ”out of thermal equilibrium”: first order electroweak phase transition:


We are not yet done: the strong CP problem

LQCD CPV = θαs

16πTr(FµνF

µν)

Neutron EDM:θ <∼ 10−9

while there is no symmetry which prohibits the θ term, so we expect θ ∼ 1.

Why is θ so small?

Peccei-Quin axion?

LQCD CPV =a(x)fa

αs

16πTr(FµνF

µν)


Hints for beyond the Standard Model physics

• Scalar fields provide a realistic (although possibly effective) description of existingparticles.

• In the presence of many fermions and gauge (vector) bosons just one scalar (Higgsboson) is unlikely. Motivation for scalar extensions of the SM:

– multi-singlet models (DM, inflation, DE)– multi-doublet models (little hierarchy problem, DM)

• Higgs boson as a tt bound state (top ”colour” dynamics required), h ∼ (tt),mt ∼ 230 GeV, W. Bardeen, Ch. Hill, and M. Lindner (stimulated by Nambu).

• Higgs boson as a pseudo-Nambu-Goldstone boson (motivated by the little hierarchyproblem), ”little Higgs models”, H.Georgi and A. Pais.

• Axion physics ∝ aF aµνF

a µν (CP and scalar fields).


Conclusions

• Yoichiro Nambu received the prize for the hypothesis of spontaneous chiralsymmetry breaking in strong interactions, with (π±, π0), as massless Nambu-Goldston bosons.

• Makoto Kobayashi and Toshihide Maskawa received the prize for discovering theorigin of CP violation (within the Glashow-Salam-Weinberg model) which hasdefended itself for 35 years as the proper source of CP violation in the quark sectorand for predicting 3 generations of quarks.


Qa5|0〉 6= 0 and Qa|0〉 = 0

⇓

• The Goldstone’s theorem:

〈0|∂µJaA µ(x)|πb(q)〉 = e−iqxfπm

2πδ

ab

Since fπ 6= 0 (Qa5|0〉 6= 0) it follows (for ∂µJa

A µ(x) = 0) that mπ = 0.

• The Goldberger-Treiman relation (satisfied within 7%):

mngA(0) = gπnfπ

where fπn pion-nucleon coupling constant and

〈n(k′)|J−A µ(x)|p(k)〉 = eiqxu(k′)[γµγ5gA(q2) + qµγ5h(q2) + · · ·

]u(k)

with q = k′ − k.


Documents

Nobel Prize 2008 - broken symmetrieshep.fuw.edu.pl/seminars/Seminaria_2008-2009/Nobel2008.pdf · at a conference at Perdue (1960) reprinted in ”Broken Symmetries, Selected Papers